¹ “Goods” will be used here in the “economic” sense: as a combination of scarce resources. The word is not meant to connote positive valuation. A “good” may not be good. Indeed, “goods” may have negative valuations placed upon them. Thus some individuals might be willing to pay to get rid of some “goods” (e.g., pollution). Similarly, “benefits” should not be viewed as necessarily carrying a positive valuation. One of the “benefits” of residing on the eastern coast of the United States of America is the breathing of industrial wastes.

² Olson, Mancur Jr., The Logic of Collective Action (New York 1968), 15, italics in original.Google Scholar

³ As quoted by Olson, n. 53, 33–34, italics added.

⁴ Waltz, Kenneth, Man, the State and War (New York 1965). 167–68Google Scholar.

⁵ Ibid., 167.

⁶ The original formulation of the problem in economic terms was by Samuelson, Paul, in “The Pure Theory of Public Expenditure,” Review of Economics and Statistics, xxxvi (November 1954), 387–89CrossRefGoogle Scholar. Other contributions to this subject include Margolis, Julius, “A Comment on the Pure Theory of Public Expenditure,” Review of Economics and Statistics, xxxvii (November 1955), 347–49CrossRefGoogle Scholar; Mancur Olson, Logic of Collective Action; and Buchanan, James, The Demand and Supply of Public Goods (Chicago 1968)Google Scholar.

⁷ This has especially been the position of Buchanan and Olson.

⁸ As Olson argues on his first two pages: “it is not in fact true that the idea that groups will act in their self-interest follows logically from the premise of rational and self-interested behavior. It does not follow, because all of the individuals in a group would gain if they achieved their group objective, that they would act to achieve that objective, even if they were rational and self-interested. Indeed, unless the number of individuals in a group is quite small, or unless there is coercion or some other special device to make the individuals act in their common interest, rational, self-interested individuals will not act to achieve their common or group interests…. The notion that groups of individuals will act to achieve their common or group interests, far from being a logical implication of the assumption that the individuals in a group will rationally further their individual interests, is in fact inconsistent with that assumption,” italics in original.

⁹ Thus, Olson concludes his book by saying: “The remaining type of group is the unorganized group-the group that has no lobby and takes no action. Groups of this kind fit the main argument of this book best of all. They illustrate its central point: that large or latent groups have no tendency voluntarily to act to further their common interests. This point was asserted in the Introduction, and it is with this point that the study must conclude. For the unorganized groups, the groups that have no lobbies and exert no pressure, are among the largest groups in the nation, and they have some of the most vital common interests.” 165.

¹⁰ We do, however, admit that with regard to the supply of many collective goods, the problems of coordination necessary to overcome the free-rider tendencies in the group could increase as the group gets larger. However, group-size is then related to the problems of free-riders by exogenous variables and not because of rationality and self-interest. As the argument below indicates, we would expect that most of the relations between size and free-riders stem from changes in the costs of the goods and the costs of communication among the individuals who would receive the goods, were they to be supplied.

¹¹ Buchanan, 91. (To paraphrase Churchill: Some small group!)

¹² It would not be necessary to belabor this point except that some of the arguments purporting to demonstrate the general occurrence of the free-rider problem have been based on crowding effects. The most obvious example of such an argument is in Buchanan, 88–90. He contrasts the probability of having a good supplied (the cost being $5,000) when 100 people each valued the good at $10 and when 10 men each valued the good at $1,000. Clearly, the contrast is misleading. The free-rider problem must be formulated so as to contrast the probability of having the good supplied when there are 10 men who value the good at $1,000 with the probability when there are 1000 men who each value the good at $1,000.

¹³ What follows is a rather technical discussion relating group size to the generation of free-rider problems. An informal sketch of the argument and its consequences are contained in Sections III and IV.

¹⁴ This assumption is a device for simplifying the exposition of the problem. All results obtained will hold without this assumption.

¹⁵ Once again, for simplicity of presentation, we assume that he considers contributing only whole units of the numeraire. If the numeraire were infinitely divisible he might consider any number of alternative donations within the range of o to n. But for a matrix presentation this would create problems. At a later stage we will discuss the consequences of relaxing this restriction.

¹⁶ The preceding footnote is relevant here also, since we begin by considering only a finite number of alternatives based on discrete donations of other members.

¹⁷ Note that i can take on values from o to n + 1. It is the columnar index.

¹⁸ The conditions governing j's choices can also be expressed in marginal terms if we allow continuous distributions of P_i and D, Then for any given level of donations, j would continue to donate so long as ndP_i—dD>o, and would stop when ndP_i — dD = o1. Of course, the second-order conditions for local maxima would also have to hold, and his utility at that point would have to be greater than when D = o or D = n. The argument is formulated in this way in Frohlich, Oppenheimer, and Young.

¹⁹ The assumption of a fixed utility derived from a given quantity of a good may be relaxed by making the utility valuations a function of the number of people who consume the good (i.e., by introducing a crowding effect). Under such a relaxation the analysis must be modified, but the modification required does not present serious problems. Note, however, that crowding can be introduced in a variety of ways. Specifying that j's utility valuation of a good is not affected by the number of other consumers fully defines a situation. If we specify that j's valuation of the good is a function of the number of other consumers, however, a specification of the exact nature of the functional relation is required. Only after that is done can crowding be handled in a formal fashion.

²⁰ We wish to thank Manfred Halpern and Ronald Rogowski for first calling our attention to this possibility.

²¹ The model developed in Frohlich, Oppenheimer, and Young is built upon these concepts. There the analysis is extended to treat such diverse subjects as the differences between political and economic entrepreneurs, political competition, politicization, the development of extremist movements, and peaking and bandwagon effects.